{"id":5991,"date":"2021-10-04T21:20:49","date_gmt":"2021-10-04T15:50:49","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5991"},"modified":"2021-10-04T21:22:21","modified_gmt":"2021-10-04T15:52:21","slug":"function-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/function-examples\/","title":{"rendered":"Function Examples"},"content":{"rendered":"

Here you will learn some function examples for better understanding of function concepts<\/a><\/span>.<\/p>\n\n\n

\n
\n\n

Example 1 : <\/span>Find the range of the given function \\(log_{\\sqrt{2}}(2-log_2(16sin^2x+1))\\)<\/p>\n

Solution : <\/span>Now 1 \\(\\le\\) \\(16sin^2x\\) + 1) \\(\\le\\) 17

\n \\(\\therefore\\)   0 \\(\\le\\) \\(log_2(16sin^2x+1)\\) \\(\\le\\) \\(log_217\\)

\n \\(\\therefore\\)   2 – \\(log_217\\) \\(\\le\\) 2 – \\(log_2(16sin^2x+1)\\) \\(\\le\\) 2

\n Now consider 0 < 2 – \\(log_2(16sin^2x+1)\\) \\(\\le\\) 2

\n \\(\\therefore\\)   -\\(\\infty\\) < \\(log_{\\sqrt{2}}(2-log_2(16sin^2x+1))\\) \\(\\le\\) \\(log_{\\sqrt{2}}2\\) = 2

\n \\(\\therefore\\)   the range is (-\\(\\infty\\), 2]<\/p>\n


\n\n \n

Example 2 : <\/span> Find the inverse of the function f(x) = \\(log_a(x + \\sqrt{(x^2+1)})\\); a > 1 and assuming it to be an onto function.<\/p>\n

Solution : <\/span>Given f(x) = \\(log_a(x + \\sqrt{(x^2+1)})\\)

\n \\(\\therefore\\)   f'(x) = \\(log_ae\\over {\\sqrt{1+x^2}}\\) > 0

\n which is strictly increasing functions.
\n Thus, f(x) is injective, given that f(x) is onto. Hence the given function f(x) is invertible.
\n Interchanging x & y

\n \\(\\implies\\)   \\(log_a(y + \\sqrt{(y^2+1)})\\) = x

\n \\(\\implies\\)   \\(y + \\sqrt{(y^2+1)}\\) = \\(a^x\\) ……..(1)

\n and   \\(\\sqrt{(y^2+1)}\\) – y = \\(a^{-x}\\) ………..(2)

\n From (1) and (2), we get y = \\(1\\over 2\\)(\\(a^x – a^{-x}\\)) or \\(f{-1}\\)(x) = \\(1\\over 2\\)(\\(a^x – a^{-x}\\)).\n <\/p>


\n\n \n

Example 3 : <\/span>Find the period of the function f(x) = \\(e^{x-[x]+|cos\\pi x|+|cos2\\pi x|+ ….. + |cosn\\pi x|}\\)<\/p>\n

Solution : <\/span>f(x) = \\(e^{x-[x]+|cos\\pi x|+|cos2\\pi x|+ ….. + |cosn\\pi x|}\\)

\n Period of x – [x] = 1

\n Period of \\(|cos\\pi x|\\) = 1

\n Period of \\(|cos2\\pi x|\\) = \\(1\\over 2\\)

\n ……………………………….

\n Period of \\(|cosn\\pi x|\\) = \\(1\\over n\\)

\n So period of f(x) will be L.C.M of all period = 1.<\/p>\n

Practice these given function examples to test your knowledge on concepts of function.<\/p>\n \n <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn some function examples for better understanding of function concepts. Example 1 : Find the range of the given function \\(log_{\\sqrt{2}}(2-log_2(16sin^2x+1))\\) Solution : Now 1 \\(\\le\\) \\(16sin^2x\\) + 1) \\(\\le\\) 17 \\(\\therefore\\)   0 \\(\\le\\) \\(log_2(16sin^2x+1)\\) \\(\\le\\) \\(log_217\\) \\(\\therefore\\)   2 – \\(log_217\\) \\(\\le\\) 2 – \\(log_2(16sin^2x+1)\\) \\(\\le\\) 2 Now consider 0 …<\/p>\n

Function Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[26],"tags":[],"yoast_head":"\nFunction Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"Solve these examples to practice and test your knowledge of function concepts.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/function-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Function Examples - 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